Jumping is the least of it, and combining it with Athletics only results in growing two skills for the price of one, and making every athlete an acrobat.

Not at all. The perk system allows for a diverging build: a guy with a 100 in "fitness" (or whatever it would wind up being called) and heavy investment in the more "athletics" side of the perk tree would be very different from the guy with a 100 in fitness and heavy investment in the "acrobatics" side of the perk tree. The new perk system changes a lot: not every mechanic has to be tied directly the the progression of a single skill anymore, allow the level of the skill itself to represent a general aptitude while the selection of perks is whatever that individual does with it.
And yes, there's a correlation between acrobatics and athletics aptitudes. A marathon runner
is going to be better at competitive gymnastics than the guy who won the nobel prize for chemistry, simply because his level of general fitness is much higher. An olympic gymnast
will beat the chessmaster in a footrace. While their areas of specialization (ie, the perks they've selected) differ, their overall ability to move their bodies with power and precision (ie, their skill level) will effect their ability to perform at other athletic-related tasks better than someone without their level of physical fitness.
Under my proposed mod, all that the levels in the skill itself would do would be to reduce the stamina costs for sprinting, climbing, and acrobatics. Without perks, in other words, you'd just become a really fit guy: you'd need to take specific perks to make yourself a champion rock climber or a gymnast or whatever.